# Definition:Cantor Set

## Definition

### As a Limit of Intersections

Define, for $n \in \N$, subsequently:

- $\map k n := \dfrac {3^n - 1} 2$

- $\displaystyle A_n := \bigcup_{i \mathop = 1}^{\map k n} \openint {\frac {2 i - 1} {3^n} } {\frac {2 i} {3^n} }$

Since $3^n$ is always odd, $\map k n$ is always an integer, and hence the union will always be perfectly defined.

Consider the closed interval $\closedint 0 1 \subset \R$.

Define:

- $\CC_n := \closedint 0 1 \setminus A_n$

The **Cantor set** $\CC$ is defined as:

- $\displaystyle \CC = \bigcap_{n \mathop = 1}^\infty \CC_n$

### From Ternary Representation

Consider the closed interval $\closedint 0 1 \subset \R$.

The **Cantor set** $\CC$ consists of all the points in $\closedint 0 1$ which can be expressed in base $3$ without using the digit $1$.

From Representation of Ternary Expansions, if any number has two different ternary representations, for example:

- $\dfrac 1 3 = 0.10000 \ldots = 0.02222$

then at most one of these can be written without any $1$'s in it.

Therefore this representation of points of $\CC$ is unique.

### As a Limit of a Decreasing Sequence

Let $\map {I_c} \R$ denote the set of all closed real intervals.

Define the mapping $t_1: \map {I_c} \R \to \map {I_c} \R$ by:

- $\map {t_1} {\closedint a b} := \closedint a {\dfrac 1 3 \paren {a + b} }$

and similarly $t_3: \map {I_c} \R \to \map {I_c} \R$ by:

- $\map {t_3} {\closedint a b} := \closedint {\dfrac 2 3 \paren {a + b} } b$

Note in particular how:

- $\map {t_1} {\closedint a b} \subseteq \closedint a b$
- $\map {t_3} {\closedint a b} \subseteq \closedint a b$

Subsequently, define inductively:

- $S_0 := \set {\closedint 0 1}$
- $S_{n + 1} := \map {t_1} {C_n} \cup \map {t_3} {C_n}$

and put, for all $n \in \N$:

- $C_n := \displaystyle \bigcup S_n$

Note that $C_{n + 1} \subseteq C_n$ for all $n \in \N$, so that this forms a decreasing sequence of sets.

Then the **Cantor set** $\mathcal C$ is defined as its limit, that is:

- $\mathcal C := \displaystyle \bigcap_{n \mathop \in \N} C_n$

These definitions are all equivalent, as shown on Equivalence of Definitions of Cantor Set.

## Also known as

Some sources refer to the **Cantor set** as **Cantor's discontinuum**.

Some sources refer to this specifically as **Cantor's middle third set** or **Cantor's ternary set**.

## Also defined as

Some sources define a **Cantor set** as any set topologically equivalent to this set.

## Also see

- Equivalence of Definitions of Cantor Set
- Cantor Space, the
**Cantor set**endowed with the Euclidean topology.

- Results about
**the Cantor set**can be found here.

## Source of Name

This entry was named for Georg Cantor.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Cantor set** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Cantor set** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Cantor set**